Average Error: 29.0 → 0.7
Time: 25.6s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r89777 = x;
        double r89778 = exp(r89777);
        double r89779 = 2.0;
        double r89780 = r89778 - r89779;
        double r89781 = -r89777;
        double r89782 = exp(r89781);
        double r89783 = r89780 + r89782;
        return r89783;
}
↓
double f(double x) {
        double r89784 = x;
        double r89785 = 2.0;
        double r89786 = pow(r89784, r89785);
        double r89787 = 0.002777777777777778;
        double r89788 = 6.0;
        double r89789 = pow(r89784, r89788);
        double r89790 = r89787 * r89789;
        double r89791 = 0.08333333333333333;
        double r89792 = 4.0;
        double r89793 = pow(r89784, r89792);
        double r89794 = r89791 * r89793;
        double r89795 = r89790 + r89794;
        double r89796 = r89786 + r89795;
        return r89796;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.0

    \[\left(e^{x} - 2\right) + e^{-x}\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.7

    \[\leadsto \color{blue}{\sqrt{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)} \cdot \sqrt{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}}\]

  5. Final simplification0.7

    \[\leadsto {x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))